How to solve q 92 on the sides and angles of a polygon.?

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2 Answers

Hence the number of sides of the convex polygon is #15# and magnitude of the angle excluded is #150^@#

Explanation:

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Let the actual number of sides of the polygon be n. So the sum of n interior angles will be #(2n-4)*90^@#
Now if #x^@# be angle excluded to give the sum of #(n-1)# angles as #2190^@# As the polygon is convex one #x<180^@#

Hence we can write

#(2n-4)xx90-x=2190#

#=>(180n-360)-x=2190#

#=>180n=(2190+360+x)#

#=>n=(2550+x)/180#

#=>n=(180*14+(30+x))/180#

#=>n=14+(30+x)/180#

As #n# is a positive integer.Here it should be greater than #14#. For #x<180^@# Considering minimum value of #(30+x)/180=1# we get #x=150^@#

Hence the number of sides of the convex polygon is #15# and magnitude of the angle excluded is #150^@#

Mar 11, 2018

#15# sides . Option C

Explanation:

A polygon can be divided into triangles by joining one vertex to all the others, The number of triangles will be:

#"sum of interior angles"/(180°)#

For example: #(1080°)/(180°) = 6 # triangles #hArr 8# sides

In this case: #(2190°)/(180°) = 12.167# triangles which is obviously not possible as the answer has to be a whole number.

The polygon is convex which means that the missing angle is not a reflex angle.

Therefore there must be #13# triangles which means #15# sides.

To find the size of the missing angle:

The sum of the interior angles will be:

#13 xx180°= 2340°#

The missing angle is #2340° -2190° = 150°#